Suppose I have a convex function $f: \mathbb{R}^n \to \mathbb{R}$. Which has a nice property: its local minimum is also the global minimum.
On the other hand, I have some constraints that limit the domain to some lattice $D \subset \mathbb{R}^n$.
E.g. $D = \mathbb{Z}^n, \mathbb{Q}^n$. or some none evenly spaced finite intervals.
And I'm wondering for what type of $D$, the local minimum = global minimum holds? And what type of $D$ such properties is lost?
In my particular example, what I have is the function and some finite intervals, if there is any way to quickly test if $f'$ with my interval $D$ has the property that local minimum is also global minimum?
Here is an example, where this property is lost: Take $D=\mathbb Z^2$, $f(x_1,x_2) = 3(2x_1-x_2)^2 + \sin(x_1)$.
For points $(x_1,x_2) \in \mathbb Z^n$ that do not satisfy $2x_1-x_2=0$, we have $f(x_1,x_2) \ge 2$. Points on the line $2x_1-x_2=0$ satisfy $f(x_1,x_2)\le 1$. So all these points are local minima, however, no global minimum exists.